Abstract
We consider a lattice model whose spins may assume a finite numberq of values. The interaction energy between two nearest-neighbor spins takes on the valueJ 1 +J 2 orJ 2, depending on whether the two spins coincide or are different but coincide modulo q1, and it is zero otherwise. This model is a generalization of the Ashkin-Teller model and exhibits the multilayer wetting phenomenon, that is, wetting by one or two or three interfacial layers, depending on the number of phases in coexistence. While we plan to consider interface properties in such a case, here we study the phase diagram of the model. We show that for large values ofq 1 andq/q 1, it exhibits, according the value ofJ 2/J 1, either a unique first-order temperature-driven phase transition at some pointβ t whereq ordered phases coexist with the disordered one, or two transition temperatures β (1)t and β (2)t , where q1 partially ordered phases coexist with the ordered ones (β (1)t ) or with the disordered one (β (2)t ), or for a particular value ofJ 2/J 1 there is a unique transition temperature where all the previous phases coexist. Proofs are based on the Pirogov-Sinai theory: we perform a random cluster representation of the model (allowing us to consider noninteger values ofq 1 andq/q 1) to which we adapt this theory.
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References
F. Dunlop, L. Laanait, A. Messager, S. Miracle-Sole, and J. Ruiz, Multilayer wetting in partially symmetric q-state models,J. Stat. Phys. 59:1383–1396 (1991).
Ya. G. Sinai,Theory of Phase Transitions: Rigorous Results (Pergamon Press, London, 1982).
J. Bricmont, K. Kuroda, and J. L. Lebowitz, First order phase transitions in lattice and continuous systems,Commun. Math. Phys. 101:501–538 (1985).
L. Laanait, A. Messager, S. Miracle-Sole, J. Ruiz, and S. Shlosman, Interfaces in the Potts model I: Pirogov-Sinai theory of the Fortuin-Kasteleyn representation,Commun. Math. Phys. 140:81–91 (1991).
C. M. Fortuin and P. W. Kasteleyn, On the random cluster model,Physica 57:536–564 (1972).
R. G. Edwards and A. D. Sokal, Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm,Phys. Rev. B 38:2009–2013 (1988).
R. H. Swendsen and J. S. Wang, Nonuniversal critical dynamics and Monte Carlo simulations,Phys. Rev. Lett. 58:86–89 (1987).
A. Benyoussef, L. Laanait, and M. Loulidi, More results on the Ashkin-Teller model, Preprint Marseille CPT-92/P.2655 (1992).
P. M. C. de Oliviera and F. C. Sa Barreto, Renormalization group studies of the Ashkin-Teller model,J. Stat. Phys. 57:53–63 (1989).
C.-E. Pfister, Phase transitions in the Ashkin-Teller model,J. Stat. Phys. 29:113–116 (1982).
R. Kotecký and D. Preiss, An inductive approach to Pirogov-Sinai theory,Supp. Rend. Circ. Matem. Palermo II(3):161–164 (1984).
N. Masaif, Étude du modèle Z(q) partiellement symétrique, Diplôme de Troisième Cycle, Faculté des Sciences, Rabat, Morocco (1992).
M. Zahradník, An alternate version of Pirogov-Sinai theory,Commun. Math. Phys. 93:559–581 (1984).
C. Borgs and J. Imbrie, A unified approach to phase diagram in field theory and statistical mechanics,Commun. Math. Phys. 123:305–328 (1989).
C.-E. Pfister, Translation invariant equilibrium states of ferromagnetic Abelian lattice systems,Commun. Math. Phys. 86:375–390 (1982).
C.-E. Pfister, On the ergodic decomposition of Gibbs random fields for ferromagnetic Abelian lattice models,Ann. N. Y. Acad. Sci. 491:170–190 (1987).
M. Zahradník, Analitycity of low temperature phase diagrams of lattice spin models,J. Stat. Phys. 47:725–755 (1987).
J. De Coninck, A. Messager, S. Miracle-Sole, and J. Ruiz, A study of perfect wetting for Potts and Blume-Capel models with correlation inequalities,J. Stat. Phys. 52:45–60 (1988).
A. Messager, S. Miracle-Sole, J. Ruiz, and S. Shlosman, Interfaces in the Potts model II: Antonov's rule and rigidity of the order disorder interface,Commun. Math. Phys. 140:275–290 (1991).
R. Kotecký, L. Laanait, A. Messager, and J. Ruiz, Theq-state Potts model in the standard Pirogov-Sinai theory: Surface tensions and Wilson loops,J. Stat. Phys. 58:199–248 (1990).
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On leave from École Normale Supérieure de Rabat, B.P. 5118, Rabat, Morocco.
On leave from Centre de Physique Théorique, CNRS, Marseille, France.
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Laanait, L., Masaif, N. & Ruiz, J. Phase coexistence in partially symmetricq-state models. J Stat Phys 72, 721–736 (1993). https://doi.org/10.1007/BF01048030
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DOI: https://doi.org/10.1007/BF01048030